透過共同層方程式統一圖神經網絡

摘要

圖神經網路通常透過特定家族的方程式來描述，這些方程式的符號掩蓋了共享的計算與結構差異。我們提出一個通用層方程式，透過七個組件來表示所涵蓋的架構：更新域、通道集、傳播算子組、逐通道訊息映射、通道融合算子、自身／殘差映射，以及更新映射。此核心分解將「資訊在何處移動」（由傳播算子組編碼）與「移動的內容」（由訊息映射編碼）加以區分。函數值填充使同一方程式得以延伸應用於局部訊息傳遞、注意力、頻譜濾波、全域通訊、關係特定通道、高階域，以及幾何訊息。我們透過對標準層的逐步化簡，以及橫跨七個非互斥架構家族的組件指派，使此統一框架明確且可驗證。一套固定的槽位規則依計算角色指派運算，並界定此框架的覆蓋邊界。此分解亦帶來組件層級的理論洞見：在端點局部訊息與節點局部更新的條件下，算子支撐限定了單層依賴關係；而在所述假設下，單層全域混合需要完整的有效算子行。由此產生的框架在共同的設計空間中整理了超過 200 種架構，支援逐組件比較與結構一致架構的生成，並將傳播選擇與過度平滑、過度擠壓、異質性及表達力聯繫起來。它進一步揭示了實證逆問題：如何將可量測的圖與任務屬性對應到經驗證的組件選擇。

English

Graph neural networks are commonly described through family-specific equations whose notation obscures shared computations and structural differences. We introduce a common layer equation that represents covered architectures through seven components: an update domain, channel set, propagation bank, per-channel message maps, channel-fusion operator, ego/residual map, and update map. The central factorization separates where information moves, encoded by the propagation bank, from what moves, encoded by the message maps. Function-valued fillings extend the same equation across local message passing, attention, spectral filtering, global communication, relation-specific channels, higher-order domains, and geometric messages. We make this unification explicit and checkable through worked reductions of canonical layers and component assignments spanning seven nonexclusive architectural families. A fixed slot discipline assigns operations by computational role and defines the framework's coverage boundary. The decomposition also yields component-level theoretical insights: under endpoint-local messages and node-local updates, operator support bounds one-layer dependencies, and one-layer global mixing requires a full effective operator row under the stated hypotheses. The resulting framework organizes more than 200 architectures in a common design space, enables component-wise comparison and generation of structurally consistent architectures, and connects propagation choices to oversmoothing, oversquashing, heterophily, and expressivity. It further exposes the empirical inverse problem of mapping measurable graph and task properties to validated component choices.